PRE CALCULUS Advanced Topics 2 — Questions and Answers
Question 1: What is the polar form of the complex number 1 + i?
- √2(cos 45° + i sin 45°) (Correct answer)
- √2(cos 90° + i sin 90°)
- 2(cos 45° + i sin 45°)
- √2(cos 135° + i sin 135°)
Correct answer: √2(cos 45° + i sin 45°)
The modulus is √(1²+1²)=√2 and the argument is arctan(1/1)=45°, giving √2(cos45°+i sin45°).
Question 2: Which conic section is represented by the equation x²/9 - y²/4 = 1?
- Hyperbola (Correct answer)
- Ellipse
- Parabola
- Circle
Correct answer: Hyperbola
The standard form x²/a² - y²/b² = 1 with opposite signs defines a hyperbola.
Question 3: What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + …?
- 3/2 (Correct answer)
- 2
- 3
- 4/3
Correct answer: 3/2
For |r|<1, S = a/(1−r) = 1/(1−1/3) = 1/(2/3) = 3/2.
Question 4: Which expression equals log₂(8) + log₂(4)?
- 5 (Correct answer)
- 7
- 12
- 32
Correct answer: 5
log₂(8)=3 and log₂(4)=2, so their sum is 5.
Question 5: The parametric equations x = 3cos(t), y = 3sin(t) describe which curve?
- A circle of radius 3 (Correct answer)
- An ellipse with semi-axes 3 and 1
- A parabola
- A line
Correct answer: A circle of radius 3
Since x²+y²=9cos²t+9sin²t=9, the curve is a circle with radius 3.
Question 6: Which of the following is a vertical asymptote of f(x) = (x+2)/((x−3)(x+1))?
- x = 3 (Correct answer)
- x = −2
- x = 2
- x = 0
Correct answer: x = 3
Vertical asymptotes occur where the denominator is zero: x=3 and x=−1.
Question 7: What is the 10th term of the arithmetic sequence 5, 8, 11, 14, …?
- 32 (Correct answer)
- 29
- 35
- 38
Correct answer: 32
The nth term is a₁+(n−1)d = 5+(10−1)(3) = 5+27 = 32.
PRE CALCULUS Advanced Topics 2
What is the polar form of the complex number 1 + i?